Multiple equilibria and indifference-threshold points in a rational addiction model

Becker and Murphy (1988) have established the existence of unstable steady states leading to threshold behavior for optimal consumption rates in intertemporal rational addiction models. In the present paper a simple linear-quadratic optimal control model is used to illustrate how their approach fits into the framework of multiple equilibria and indifference-threshold points. By changing the degree of addiction and the level of harmfulness we obtain a variety of behavioral patterns.
In particular we show that when the good is harmful as well as very addictive, an indifference-threshold point, also known in literature as Skiba point, separates patterns of converging to either zero or maximal consumption, where the latter occurs in case of a high level of past consumption. This implicitly shows that an individual needs to be aware in time of these characteristics of the good. Otherwise, he/she may start consuming so much that in the end he/she is totally addicted.